我是一名数学老师，请将圆转化成长方形的过程用动画展示，并讲解，最后推导出公式---**Textual Information:** * **Activity/Question 8:** 动 8 把第 117 页上半部分的圆剪下来，按 16 等份剪开，再拼一拼，看看能拼成什么图形。 (Activity 8: Cut out the circle from the upper half of page 117, cut it into 16 equal parts, and then put them together to see what shape you can make.) * **Annotation (Speech bubble 1):** 拼成了一个近似的平行四边形。 (Put together to form an approximate parallelogram.) * **Follow-up Question:** 如果把圆平均分成 32 份、64 份……拼成的图形会有什么变化？ (If the circle is divided into 32 parts, 64 parts... what changes will occur in the shape put together?) * **Annotation (Speech bubble 2):** 平均分的份数越多，拼成的图形越接近长方形。 (The more equal parts it is divided into, the closer the shape put together approaches a rectangle.) * **Question (Speech bubble 3):** 拼成的长方形与原来的圆有什么关系？ (What is the relationship between the rectangle put together and the original circle?) * **Statement (Box 1):** 长方形的面积与圆的面积相等。 (The area of the rectangle is equal to the area of the circle.) * **Statement (Box 2):** 长方形的宽是圆的半径。 (The width of the rectangle is the radius of the circle.) * **Statement (Box 3):** 长方形的长是圆周长的一半。 (The length of the rectangle is half of the circle's circumference.) * **Question:** 如果圆的半径是 r, 这个长方形的长和宽各应怎样表示？在小组里说说，根据长方形的面积计算方法怎样计算圆的面积。 (If the radius of the circle is r, how should the length and width of this rectangle be represented? Discuss in groups how to calculate the area of a circle based on the method for calculating the area of a rectangle.) * **Mathematical Formula Derivation:** * 长方形的面积 = 长 × 宽 (Area of rectangle = length × width) * 圆的面积 = πr × r (Area of circle = πr × r) * = πr² * **Final Formula Statement:** 如果用 S 表示圆的面积，上面的公式可以写成： (If S represents the area of the circle, the above formula can be written as:) * S = πr² * **Top Right Corner Label:** 圆 (Circle) **Chart/Diagram Description:** The image contains two sets of diagrams illustrating the method of transforming a circle into a shape to derive its area formula. 1. **First Diagram Set:** * **Elements:** A circle divided into 16 equal sectors (8 yellow, 8 pink). An arrow pointing right. A shape formed by arranging the 16 sectors side-by-side, with bases alternating up and down, forming a shape resembling a parallelogram with curved top and bottom edges. * **Labels:** A speech bubble next to the rearranged shape stating it's an "approximate parallelogram". 2. **Second Diagram Set:** * **Elements:** A circle divided into a much larger number of equal sectors (many yellow and pink). An arrow pointing right. A shape formed by arranging these sectors side-by-side, similar to the first set, but now resembling a rectangle with slightly wavy top and bottom edges. * **Labels:** * An arrow pointing vertically along the right side of the rearranged shape, labeled "高 (即 r)" (Height (i.e., r)). * A double-headed arrow along the bottom base of the rearranged shape, labeled "C/2 (即 πr)" (C/2 (i.e., πr)). * A speech bubble below this diagram set stating that with more divisions, the shape approaches a rectangle. The diagrams visually show how dividing a circle into increasingly many sectors and rearranging them can approximate a rectangle whose area is equal to the circle's area. The dimensions of the approximate rectangle are labeled in the second diagram, relating them to the circle's radius (r) and circumference (C).